Materials modelling using density functional theory properties and predictions pdf
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- PDF Download Materials Modelling using Density Functional Theory: Properties and Predictions
- Materials Modelling Using Density Functional Theory: Properties and Predictions
- Materials Modelling using Density Functional Theory (eBook, PDF)
- Materials Modelling using Density Functional Theory (eBook, PDF)
PDF Download Materials Modelling using Density Functional Theory: Properties and Predictions
This book serves two purposes: 1 to provide worked examples of using DFT to model materials properties, and 2 to provide references to more advanced treatments of these topics in the literature. It is not a definitive reference on density functional theory. Along the way to learning how to perform the calculations, you will learn how to analyze the data, make plots, and how to interpret the results. This book is very much "recipe" oriented, with the intention of giving you enough information and knowledge to start your research.
In that sense, many of the computations are not publication quality with respect to convergence of calculation parameters. You will read a lot of python code in this book. I believe that computational work should always be scripted. Scripting provides a written record of everything you have done, making it more probable you or others could reproduce your results or report the method of its execution exactly at a later time.
Similar code would be used for other calculators, e. GPAW, Jacapo, etc… you would just have to import the python modules for those codes, and replace the code that defines the calculator. A comprehensive overview of DFT is beyond the scope of this book, as excellent reviews on these subjects are readily found in the literature, and are suggested reading in the following paragraph. Instead, this chapter is intended to provide a useful starting point for a non-expert to begin learning about and using DFT in the manner used in this book.
Much of the information presented here is standard knowledge among experts, but a consequence of this is that it is rarely discussed in current papers in the literature. A secondary goal of this chapter is to provide new users with a path through the extensive literature available and to point out potential difficulties and pitfalls in these calculations.
A modern and practical introduction to density functional theory can be found in Sholl and Steckel sholldensit-funct-theor. The Chemist's Guide to DFT koch is more readable and contains more practical information for running calculations, but both of these books focus on molecular systems.
The standard texts in solid state physics are by Kittel kittel and Ashcroft and Mermin ashcroft-mermin. Both have their fine points, the former being more mathematically rigorous and the latter more readable. However, neither of these books is particularly easy to relate to chemistry. For this, one should consult the exceptionally clear writings of Roald Hoffman hoffmann , RevModPhys. In this chapter, only the elements of DFT that are relevant to this work will be discussed. An excellent review on other implementations of DFT can be found in Reference freemandensit , and details on the various algorithms used in DFT codes can be found in Refs.
One of the most useful sources of information has been the dissertations of other students, perhaps because the difficulties they faced in learning the material are still fresh in their minds. The Ph.
Finally, another excellent overview of DFT and its applications to bimetallic alloy phase diagrams and surface reactivity is presented in the PhD thesis of Robin Hirschl hirschlbinar-trans-metal-alloy-their-surfac. He later shared the Nobel prize with Paul A. Dirac in for this discovery. Even if this rough estimate is off by an order of magnitude, a system with electrons is still very small, for example, two Ru atoms if all the electrons are counted, or perhaps ten Pt atoms if only the valence electrons are counted.
Thus, the wave function method, which has been extremely successful in studying the properties of small molecules, is unsuitable for studies of large, extended solids. Interestingly, this difficulty was recognized by Dirac as early as , when he wrote "The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the application of these laws leads to equations much too complicated to be soluble.
In , Hohenberg and Kohn showed that the ground state total energy of a system of interacting electrons is a unique functional of the electron density PhysRev. By definition, a function returns a number when given a number. A functional returns a number when given a function. Hohenberg and Kohn further identified a variational principle that appeared to reduce the problem of finding the ground state energy of an electron gas in an external potential i.
Unfortunately, the definition of the functional involved a set of 3N-dimensional trial wave functions. In , Kohn and Sham made a significant breakthrough when they showed that the problem of many interacting electrons in an external potential can be mapped exactly to a set of noninteracting electrons in an effective external potential PhysRev.
This led to a set of self-consistent, single particle equations known as the Kohn-Sham KS equations:. Thus, the density needs to be known in order to define the effective potential so that Eq. To solve Eq. Finally, the ground state energy is given by:. The other half of the prize went to John Pople for his efforts in wave function based quantum mechanical methods RevModPhys. Provided the exchange-correlation energy functional is known, Eq.
However, the exact form of the exchange-correlation energy functional is not known, thus approximations for this functional must be used. It is exact for a uniform electron gas, and is anticipated to be a reasonable approximation for slowly varying densities. In molecules and solids, however, the density tends to vary substantially in space.
Despite this, the LDA has been very successfully used in many systems. It tends to predict overbonding in both molecular and solid systems fuchspseud , and it tends to make semiconductor systems too metallic the band gap problem perdewelect-kohn-sham.
The generalized gradient approximation includes corrections for gradients in the electron density, and is often implemented as a corrective function of the LDA. The form of this corrective function, or "exchange enhancement" function determines which functional it is, e. Finally, there are increasingly new types of functionals in the literature. The so-called hybrid functionals, such as B3LYP, are more popular with gaussian basis sets e.
None of these other types of functionals were used in this work. For more details see Chapter 6 in Ref. In a periodic solid, one can use Bloch's theorem to show that the wave function for an electron can be expressed as the product of a planewave and a function with the periodicity of the lattice ashcroft-mermin :.
Bloch's theorem sets the stage for using planewaves as a basis set, because it suggests a planewave character of the wave function. This also converts Eq. In aperiodic systems, such as systems with even one defect, or randomly ordered alloys, there is no periodic unit cell. Instead one must represent the portion of the system of interest in a supercell, which is then subjected to the periodic boundary conditions so that a planewave basis set can be used.
It then becomes necessary to ensure the supercell is large enough to avoid interactions between the defects in neighboring supercells. The case of the randomly ordered alloy is virtually hopeless as the energy of different configurations will fluctuate statistically about an average value. These systems were not considered in this work, and for more detailed discussions the reader is referred to Ref.
Once a supercell is chosen, however, Bloch's theorem can be applied to the new artificially periodic system. To get a perfect expansion, one needs an infinite number of planewaves.
Luckily, the coefficients of the planewaves must go to zero for high energy planewaves, otherwise the energy of the wave function would go to infinity. This provides justification for truncating the planewave basis set above a cutoff energy. Careful testing of the effect of the cutoff energy on the total energy can be done to determine a suitable cutoff energy. The cutoff energy required to obtain a particular convergence precision is also element dependent, shown in Table tab:pwcut.
It can also vary with the "softness" of the pseudopotential. Thus, careful testing should be done to ensure the desired level of convergence of properties in different systems. Table tab:pwcut refers to convergence of total energies. These energies are rarely considered directly, it is usually differences in energy that are important.
These tend to converge with the planewave cutoff energy much more quickly than total energies, due to cancellations of convergence errors. In this work, eV was found to be suitable for the H adsorption calculations, but a cutoff energy of eV was required for O adsorption calculations. Bloch's theorem eliminates the need to calculate an infinite number of wave functions, because there are only a finite number of electrons in the unit super cell.
However, there are still an infinite number of discrete k points that must be considered, and the energy of the unit cell is calculated as an integral over these points. It turns out that wave functions at k points that are close together are similar, thus an interpolation scheme can be used with a finite number of k points. This also converts the integral used to determine the energy into a sum over the k points, which are suitably weighted to account for the finite number of them.
There will be errors in the total energy associated with the finite number of k, but these can be reduced and tested for convergence by using higher k-point densities. An excellent discussion of this for aperiodic systems can be found in Ref.
The use of these k point setups amounts to an expansion of the periodic function in reciprocal space, which allows a straight-forward interpolation of the function between the points that is more accurate than with other k point generation schemes PhysRevB.
The core electrons of an atom are computationally expensive with planewave basis sets because they are highly localized. This means that a very large number of planewaves are required to expand their wave functions. Furthermore, the contributions of the core electrons to bonding compared to those of the valence electrons is usually negligible.
In fact, the primary role of the core electron wave functions is to ensure proper orthogonality between the valence electrons and core states. Consequently, it is desirable to replace the atomic potential due to the core electrons with a pseudopotential that has the same effect on the valence electrons PhysRevB. There are essentially two kinds of pseudopotentials, norm-conserving soft pseudopotentials PhysRevB.
In either case, the pseudopotential function is generated from an all-electron calculation of an atom in some reference state. In norm-conserving pseudopotentials, the charge enclosed in the pseudopotential region is the same as that enclosed by the same space in an all-electron calculation.
In ultrasoft pseudopotentials, this requirement is relaxed and charge augmentation functions are used to make up the difference. As its name implies, this allows a "softer" pseudopotential to be generated, which means fewer planewaves are required to expand it. The pseudopotentials are not unique, and calculated properties depend on them. However, there are standard methods for ensuring the quality and transferability to different chemical environments of the pseudopotentials PhysRevB.
At absolute zero, the occupancies of the bands of a system are well-defined step functions; all bands up to the Fermi level are occupied, and all bands above the Fermi level are unoccupied. There is a particular difficulty in the calculation of the electronic structures of metals compared to semiconductors and molecules.
In molecules and semiconductors, there is a clear energy gap between the occupied states and unoccupied states. Thus, the occupancies are insensitive to changes in the energy that occur during the self-consistency cycles.
In metals, however, the density of states is continuous at the Fermi level, and there are typically a substantial number of states that are close in energy to the Fermi level. Consequently, small changes in the energy can dramatically change the occupation numbers, resulting in instabilities that make it difficult to converge to the occupation step function.
A related problem is that the Brillouin zone integral which in practice is performed as a sum over a finite number of k points that defines the band energy converges very slowly with the number of k points due to the discontinuity in occupancies in a continuous distribution of states for metals gillancalcul , Kresse
Materials Modelling Using Density Functional Theory: Properties and Predictions
Density-functional theory DFT is a computational quantum mechanical modelling method used in physics , chemistry and materials science to investigate the electronic structure or nuclear structure principally the ground state of many-body systems , in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals , i. In the case of DFT, these are functionals of the spatially dependent electron density. DFT is among the most popular and versatile methods available in condensed-matter physics , computational physics , and computational chemistry. DFT has been very popular for calculations in solid-state physics since the s. However, DFT was not considered accurate enough for calculations in quantum chemistry until the s, when the approximations used in the theory were greatly refined to better model the exchange and correlation interactions.
A test set of chemically and topologically diverse Metal—Organic Frameworks MOFs with high accuracy experimentally derived crystallographic structure data was compiled. The test set was also used to assess the variance in performance of DFT functionals for elastic properties and atomic partial charges. The DFT predicted elastic properties such as minimum shear modulus and Young's modulus can differ by an average of 3 and 9 GPa for rigid MOFs such as those in the test set. We find that while there are differences in the magnitude of the properties predicted by the various functionals, these discrepancies are small compared to the accuracy necessary for most practical applications. If you are not the author of this article and you wish to reproduce material from it in a third party non-RSC publication you must formally request permission using Copyright Clearance Center.
Materials Modelling using Density Functional Theory (eBook, PDF)
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Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Jain and Yongwoo Shin and K. Jain , Yongwoo Shin , K.
This book is an introduction to the quantum theory of materials and first-principles computational materials modelling. It explains how to use density functional theory as a practical tool for calculating the properties of materials without using any empirical parameters. This book is intended for senior undergraduate and first-year graduate students in materials science, physics, chemistry, and engineering who are approaching for the first time the study of materials at the atomic scale.
It explains how to use density functional theory as a practical tool for calculating the properties of materials without using any empirical parameters. This book is intended for senior undergraduate and first-year graduate students in materials science, physics, chemistry, and engineering who are approaching for the first time the study of materials at the atomic scale.
Materials Modelling using Density Functional Theory (eBook, PDF)
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This book serves two purposes: 1 to provide worked examples of using DFT to model materials properties, and 2 to provide references to more advanced treatments of these topics in the literature. It is not a definitive reference on density functional theory. Along the way to learning how to perform the calculations, you will learn how to analyze the data, make plots, and how to interpret the results. This book is very much "recipe" oriented, with the intention of giving you enough information and knowledge to start your research. In that sense, many of the computations are not publication quality with respect to convergence of calculation parameters. You will read a lot of python code in this book.